The study of correlation among different scattering parameters in an aggregate dust model

Abstract

We study the light scattering properties of aggregate particles in a wide range of complex refractive indices ($m = n + i k$, where $1.4 \le n \le 2.0$, $0.001 \le k \le1.0$) and wavelengths ($0.45 \le \lambda\le1.25 \mbox{ }\upmu \mbox{m}$) to investigate the correlation among different parameters e.g., the positive polarization maximum ($P_{\mathrm{max}}$), the amplitude of the negative polarization ($P_{\mathrm{min}}$), geometric albedo ($A$), $(n,k)$ and $\lambda$. Numerical computations are performed by the Superposition T-matrix code with Ballistic Cluster–Cluster Aggregate (BCCA) particles of 128 monomers and Ballistic Aggregates (BA) particles of 512 monomers, where monomer’s radius of aggregates is considered to be 0.1 μm. At a fixed value of $k$, $P_{\mathrm{max}}$ and $n$ are correlated via a quadratic regression equation and this nature is observed at all wavelengths. Further, $P_{\mathrm{max}}$ and $k$ are found to be related via a polynomial regression equation when $n$ is taken to be fixed. The degree of the equation depends on the wavelength, higher the wavelength lower is the degree. We find that $A$ and $P_{\mathrm{max}}$ are correlated via a cubic regression at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ whereas this correlation is quadratic at higher wavelengths. We notice that $|P_{\mathrm{min}}|$ increases with the decrease of $P_{\mathrm{max}}$ and a strong linear correlation between them is observed when $n$ is fixed at some value and $k$ is changed from higher to lower value. Further, at a fix value of $k$, $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ can be fitted well via a quartic regression equation when $n$ is changed from higher to lower value. We also find that $P_{\mathrm{max}}$ increases with $\lambda$ and they are correlated via a quartic regression.

1 Introduction

The light scattering properties of aggregated particles have been extensively studied through the use of various numerical techniques. Among them, the Superposition T-matrix (STM) code and the Discrete Dipole Approximation (DDA) code are widely used by researchers. In T-matrix approach, $N$ spheres is represented by the superposition of fields scattered from each of the spheres in the ensemble (Mackowski and Mishchenko 1996). This technique relates a superposition solution to Maxwell’s equations for the multiple spherical boundary domain where one can study the light scattering properties of aggregate particles in either fixed or random orientations. In DDA approach, the target is replaced by an array of $N$ point dipoles where the spacing between the dipoles is taken to be small comparable to the wavelength. The incident periodic wave interacting with this array of point dipoles can be solved exactly by DDA code (Draine and Flatau 1994). The calculation with T-matrix is fast because the orientation average of scattering matrix can be performed analytically.

Many investigators studied the light scattering properties of cometary dust coma using above two methods (Petrova et al. 2004; Kimura et al. 2006; Lasue and Levasseur-Regourd 2006; Bertini et al. 2007; Das et al. 2008a, 2008b). These papers mainly investigated the angular and spectral dependencies of polarization and intensity for aggregate particles consisting of identical homogeneous spheres. These studies nicely portrayed the dependence of polarization and intensity on different parameters such as composition, size, and number of constituent spherical particles, structure and porosity of the cluster. It is well accepted by scientific community that interstellar grains are composites of many small subunits with cavities and voids (Mathis and Whiffen 1989; Ossenkopf 1993; Iati et al. 2004). Iati et al. (2004) calculated the relevant optical properties of cosmic dust grains of amorphous carbon and astronomical silicates using aggregate dust model. Graham et al. (2007) used the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) to make the first polarization maps of the AU Microscopii debris disk. They modeled the observed data and suggested that very porous ($P \approx0.91\mbox{--}0.94$) micron sized spherical grains or aggregates can produce the observed features based on Mie theory and Superposition T-matrix theory for Ballistic Aggregates (BA) clusters.

In this paper, we study the light scattering properties of aggregate particles in a wide range of complex refractive indices ($m = n + i k$, where $1.4 \le n \le 2.0$, $0.001 \le k \le1.0$) and wavelengths ($0.45 \le\lambda\le1.25 \mbox{ }\upmu \mbox{m}$) to investigate the correlation among different parameters. The structure of the paper is as follows: in Sect.  2 we describe the aggregate dust model, in Sect.  3 the results obtained from numerical simulations are presented with suitable interpretations, in Sect.  4 results obtained from correlation equations are shown, and a set of conclusions based on this work is finally presented in the last section.

2 Aggregate dust model

In our simulations, the aggregates are built using the ballistic aggregation procedure (Meakin 1983, 1984). The aggregates can be created either via single-particle aggregation or through cluster–cluster aggregation. Using Monte Carlo simulation, Ballistic Particle–Cluster Aggregate (BPCA) structure has been built by allowing single particles to join the cluster of the particle, whereas Ballistic Cluster–Cluster Aggregate (BCCA) structure has been created by allowing clusters of particles to stick together. It is to be noted that BCCA is more porous than BPCA. The fractal dimensions of BPCA and BCCA particles are given by $D \thickapprox3$ and $\thickapprox2$, respectively (Meakin 1984). In this paper, computations have been performed with BCCA structure having $N = 128$.

We have also considered the Ballistic Aggregates (BA) constructed by Shen et al. (2008) in our simulations to study the correlation among different scattering parameters in an aggregate dust model. It is to be noted that the BA structure is generated by Ballistic Agglomeration process having porosity $\approx0.87$ and fractal dimension $\approx3.0$, which is similar to the Ballistic Particle Cluster Aggregate (BPCA). In our computation, we considered BA with $N = 512$ having porosity $\approx0.87$.

Greenberg and Hage (1990) first suggested the presence of 0.1 μm grains as basic constituents of comet dust aggregates to explain the observed emission band feature of comet 1P/Halley. Later, many investigators modeled the light scattering properties of comet dust considering monomer radius of 0.1 μm (Kimura et al. 2006; Das et al. 2008a, 2008b, 2011). In our present work, we have also considered the radius of monomer to be 0.1 μm so that our results can be compared with previous work.

The radius of an aggregate particle is given by $a_{v} = a_{m}N^{1/3}$, where $a_{m}$ is the radius of monomer and $N$ is the number of monomer. The size parameter of the monomer is given by $x = 2\pi a_{m}/\lambda$, where $\lambda$ is the wavelength of incident radiation. In this paper, we consider five wavelengths from optical to near infrared which are given by 0.45, 0.65, 0.85, 1.05 and 1.25 μm whose $x$ are 1.397, 0.967, 0.740, 0.599 and 0.503. In this work, BCCA structure with 128 monomers, having porosity 0.94, is considered throughout the computations. Thus, $a_{v} \approx0.5 \mbox{ }\upmu \mbox{m}$ for BCCA structure with $N = 128$ and $a_{v} \approx0.8 \mbox{ }\upmu \mbox{m}$ for BA structure with $N = 512$. Further, the characteristic radius ($R = 2.0\times a_{v}$) of BA (512) structure, defined by (Shen et al. 2008), is given by 1.6 μm whereas $R$ ($\approx2.2\times a_{v}$) for BCCA structure is given by 1.1 μm.

We execute our calculations using parallel Multi-sphere T-matrix code version 3.0 developed by Mackowski and Mishchenko (2013). This code is highly efficient in the parallel computational environment and could be used to study the light scattering properties of aggregates.

In our work, we study the positive polarization maximum ($P_{\mathrm{max}}$), amplitude of the negative polarization ($P_{\mathrm{min}}$) and geometric albedo ($A$) for aggregate particles with a wide range of complex refractive indices ($1.4 \le n \le 2.0$, $0.001 \le k \le1.0$) and wavelengths ($0.45 \le\lambda\le1.25 \mbox{ }\upmu \mbox{m}$). The geometric albedo ($A$) can be calculated from the definition provided by Hanner et al. (1981), which is given by $[{S}_{11}(180^{\circ})~\lambda ^{2}]/(4\pi G)$. Here ${S}_{11}(180^{\circ})$ is the first element of the scattering matrix at the exact back scattering direction and $G$ is the geometric cross section of aggregates.

3 Results

3.1 Correlation between polarization maximum and complex refractive index

3.1.1 $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

To study the dependence of $n$ and $k$ on polarization maximum ($P_{\mathrm{max}}$), we can either plot $P_{\mathrm{max}}$ versus $k$ by keeping $n$ fixed or plot $P_{\mathrm{max}}$ versus $n$ by keeping $k$ fixed. We first plot $P_{\mathrm{max}}$ versus $k$ for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$ respectively, in a single frame, for BCCA structure with $N= 128$. The plot is shown in Fig. 1, where $R > \lambda$. It can be seen from figure that if $k$ is fixed at any value between 0.001 and 1.0, $P_{\mathrm{max}}$ decreases with increase of $n$ from 1.4 to 2.0. The vertical range of $P_{\mathrm{max}}$ also decreases when $k$ increases. This range is maximum at $k = 0.001$ (where $P_{\mathrm{max}} = [0.068, 0.845]$) and minimum at $k = 1.0$ (where $P_{\mathrm{max}} = [0.571, 0.746]$). The slope of $P_{\mathrm{max}}$ versus $k$ increases with increase of $n$ from 1.4 to 2.0. When $n \le1.6$, the change of $P_{\mathrm{max}}$ with $k$ is noted to be small. Further, variation of $P_{\mathrm{max}}$ is almost independent of $k$ when it is ${\ge}0.5$, for all values of $n$.

Fig. 1

Polarization maximum ($P_{\mathrm{max}}$) is plotted against imaginary part of refractive index ($k$) for real part of refractive indices ($n$) = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0 respectively. The simulations are done for BCCA structure with $N = 128$. Here, the radius of monomer ($a_{m}$) and the wavelength of incident radiation ($\lambda$) are fixed at 0.1 μm and 0.45 μm. The best fit curves correspond to quartic/biquadratic regression of the form $P_{\mathrm{max}} = A_{n} k^{4} + B_{n} k^{3} + C_{n} k^{2} + D_{n} k + E_{n}$, which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of equation at different values of $n$ are shown in Table 1

We have noted that $P_{\mathrm{max}}$ and $k$ can be fitted by quartic/biquadratic regression where coefficient of determination ¹ ($R^{2}$) for each equation is ≈0.99. The best fit equation is given by

$$ P_{\mathrm{max}} = A_{n} k^{4} + B_{n} k^{3} + C_{n} k^{2} + D_{n} k + E_{n} , $$

(1a)

where $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ and $E_{n}$ are $n$-dependent coefficients of (1a) (where $1.4 \le n \le2.0$, $0.001 \le k \le1$). The coefficients obtained for different values of $n$ are shown in Table 1. If we plot coefficients $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ and $E_{n}$ versus $n$ (shown in Fig. 2), it can be seen that the best fit curves correspond to quintic regression, which have ${R^{2} \approx 0.99}$.

Fig. 2

Coefficients $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ and $E_{n}$ are plotted against the real part of refractive index ($n$) (see Table 1). The best fit curves correspond to quintic regression shown in (1b)–(1f), which have $R^{2} \approx 0.99$. All coefficients of (1b)–(1f) are shown in Table 2

Table 1 All co-efficients of (1a) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

Thus coefficients are given by

$$\begin{aligned}& A_{n} = \alpha_{1} n^{5} + \alpha_{2} n^{4} + \alpha_{3} n^{3} + \alpha_{4} n^{2} + \alpha_{5} n + \alpha_{6} , \end{aligned}$$

(1b)

$$\begin{aligned}& B_{n} = \beta_{1} n^{5} + \beta_{2} n^{4} + \beta_{3} n^{3} + \beta_{4} n^{2} + \beta_{5} n +\beta_{6} , \end{aligned}$$

(1c)

$$\begin{aligned}& C_{n} = \gamma_{1} n^{5} + \gamma_{2} n^{4} + \gamma_{3} n^{3} + \gamma_{4} n^{2} + \gamma_{5} n + \gamma_{6} , \end{aligned}$$

(1d)

$$\begin{aligned}& D_{n} = \delta_{1} n^{5} + \delta_{2} n^{4} + \delta_{3} n^{3} + \delta_{4} n^{2} + \delta_{5} n + \delta_{6} , \end{aligned}$$

(1e)

$$\begin{aligned}& E_{n} = \epsilon_{1} n^{5} + \epsilon_{2} n^{4} + \epsilon_{3} n^{3} + \epsilon_{4} n^{2} + \epsilon_{5} n + \epsilon_{6} . \end{aligned}$$

(1f)

All coefficients of (1b)–(1f) are shown in Table 2. Thus it can be seen that for any selected values of $n$ and $k$, $P_{\mathrm{max}}$ can be estimated using the relations (1a) and (1b)–(1f).

Table 2 All co-efficients of (1b)–(1f) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We have also executed the computations for BA/BPCA structure with 512 monomers at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ to see whether a similar correlation exists or not at a high number of monomers (i.e., at large size of the aggregate). We plot $P_{\mathrm{max}}$ versus $k$ for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$ respectively, which is shown in Fig. 3. It is interesting to notice that they can also be fitted by quartic/biquadratic regression where $R^{2} \approx 0.99$. This shows that nature is similar even if the number of the monomer is high, but the coefficients of equations obtained in this case are different from the coefficients obtained for BCCA with $N = 128$. We did not show any equation in this case.

Fig. 3

Same as Fig. 1, but for BA structure with $N = 512$

We now plot $P_{\mathrm{max}}$ against $n$ for $k = 0.001, 0.005, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1$. The plots are shown in Fig. 4(i) and (ii) for $N = 128$. It can be seen from figure that $P_{\mathrm{max}}$ decreases as $n$ increases, for all value of $k$. This tendency is less prominent for aggregate particles when $k$ is high. For low values of $n$, the variation of $P_{\mathrm{max}}$ is found to be less pronounced for increasing values of $k$ from 0.001 to 1.0. It is observed that $P_{\mathrm{max}}$ and $n$ can be fitted by quadratic regression, where $R^{2}$ for each equation is ≈0.99, which is given by

$$ P_{\mathrm{max}} = K_{k} n^{2} + L_{k} n + M_{k}, $$

(2a)

where $K_{k}$, $L_{k}$ and $M_{k}$ are $k$-dependent coefficients of (2a). The coefficients obtained for different values of $k$ are shown in Table 3. If we plot coefficients $K_{k}$, $L_{k}$ and $M_{k}$ versus $k$ (shown in Fig. 5), it can be seen that the best fit curves correspond to sextic/hexic regression, which have ${R^{2} \approx 0.99}$.

Fig. 4

Polarization maximum ($P_{\mathrm{max}}$) is plotted against the real part of refractive index ($n$) for the imaginary part of refractive indices ($k$) = 0.001, 0.005, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8 and 1, respectively. The simulations are done for BCCA structure with $N = 128$ and $a_{m} = 0.1\mbox{ }\upmu \mbox{m}$, where $\lambda$ is fixed at 0.45 μm. The best fit curves correspond to quadratic regression of the form $P_{\mathrm{max}} = K_{k} n^{2} + L_{k} n + M_{k}$, which have $R^{2}\approx 0.99$. All coefficients of equation at different values of $k$ are shown in Table 3

Fig. 5

Coefficients $K_{k}$, $L_{k}$ and $M_{k}$ are plotted against the imaginary part of refractive index ($k$) (see Table 3). The best fit curves correspond to sextic/hexic regression shown in (2b)–(2d), which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of (2b)–(2d) are shown in Table 4

Table 3 All co-efficients of (2a) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

Thus coefficients are given by

$$\begin{aligned}& K_{k} =\epsilon_{1} k^{6} + \epsilon_{2} k^{5} + \epsilon_{3} k^{4} + \epsilon_{4} k^{3} + \epsilon_{5} k^{2} + \epsilon_{6} k + \epsilon_{7}, \end{aligned}$$

(2b)

$$\begin{aligned}& L_{k} =\zeta_{1} k^{6} + \zeta_{2} k^{5} + \zeta_{3} k^{4} + \zeta_{4} k^{3} + \zeta_{5} k^{2} + \zeta_{6} k + \zeta_{7}, \end{aligned}$$

(2c)

$$\begin{aligned}& M_{k} =\eta_{1} k^{6} + \eta_{2} k^{5} + \eta_{3} k^{4} + \eta_{4} k^{3} + \eta _{5} k^{2} + \eta_{6} k + \eta_{7}. \end{aligned}$$

(2d)

All coefficients of (2b)–(2d) are shown in Table 4. Thus it can be seen that $P_{\mathrm{max}}$ can be also estimated using (2a) and (2b)–(2d).

Table 4 All co-efficients of (2b)–(2d) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We now study the correlation between $P_{\mathrm{max}}$ and $n$ for BA/BPCA structure with $N = 512$. A strong correlation between them is found which is related via a quadratic regression, shown in Fig. 6. Nature is again similar to that of BCCA structure ($N = 128$).

Fig. 6

Same as Fig. 4, but for BA structure with $N = 512$

3.1.2 $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

We study the correlation between $P_{\mathrm{max}}$ and $(n, k)$ at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. We plot $P_{\mathrm{max}}$ versus $k$ for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$ respectively, The plot is shown in Fig. 7. The nature of curves is almost same as that of Fig. 1, but the best fit curves correspond to cubic regression of the form:

$$ P_{\mathrm{max}} = A_{n} k^{3} + B_{n} k^{2} + C_{n} k + D_{n}, $$

(3a)

where $A_{n}$, $B_{n}$, $C_{n}$ and $D_{n}$ are $n$-dependent coefficients of (3a). The coefficients obtained for different values of $n$ are shown in Table 5. The plots of $A_{n}$, $B_{n}$, $C_{n}$ and $D_{n}$ versus $n$ in Fig. 8 show that the best fit curves correspond to quintic regression, which have $R^{2}\approx 0.99$. The degree of following non-linear equations is same as that of (1b)–(1f).

Fig. 7

The same as Fig. 1, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The simulations are done for BCCA structure with $N = 128$. The best fit curves correspond to cubic regression of the form $P_{\mathrm{max}} = A_{n} k^{3} + B_{n} k^{2} + C_{n} k + D_{n}$, which have $R^{2}\approx 0.99$. All coefficients of equation at different values of $n$ are shown in Table 5

Fig. 8

The same as Fig. 2, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The best fit curves correspond to quintic regression shown in (3b)–(3e), which have $R^{2}\approx0.99$. All coefficients of (3b)–(3e) are shown in Table 6

Table 5 All co-efficients of (3a) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

Thus coefficients are given by

$$\begin{aligned}& A_{n} = \alpha_{1} n^{5} + \alpha_{2} n^{4} + \alpha_{3} n^{3} + \alpha_{4} n^{2} + \alpha_{5} n + \alpha_{6} , \end{aligned}$$

(3b)

$$\begin{aligned}& B_{n} = \beta_{1} n^{5} + \beta_{2} n^{4} + \beta_{3} n^{3} + \beta_{4} n^{2} + \beta_{5} n +\beta_{6} , \end{aligned}$$

(3c)

$$\begin{aligned}& C_{n} = \gamma_{1} n^{5} + \gamma_{2} n^{4} + \gamma_{3} n^{3} + \gamma_{4} n^{2} + \gamma_{5} n + \gamma_{6} , \end{aligned}$$

(3d)

$$\begin{aligned}& D_{n} = \delta_{1} n^{5} + \delta_{2} n^{4} + \delta_{3} n^{3} + \delta_{4} n^{2} + \delta_{5} n + \delta_{6}. \end{aligned}$$

(3e)

All coefficients of (3b)–(3e) are shown in Table 6.

Table 6 All co-efficients of (3b)–(3e) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

We now plot $P_{\mathrm{max}}$ against $n$ for $k = 0.001, 0.005, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1$, which is shown in Fig. 9(i) and (ii). The nature of curves is similar to that of Fig. 4, i.e., $P_{\mathrm{max}}$ decreases as $n$ increases, for all value of $k$. It is noted that $P_{\mathrm{max}}$ and $n$ can be fitted by quadratic regression, where $R^{2}$ for each equation is ≈0.99, which is given by

$$ P_{\mathrm{max}} = K_{k} n^{2} + L_{k} n + M_{k}, $$

(4a)

where $K_{k}$, $L_{k}$ and $M_{k}$ are $k$-dependent coefficients of (4a). The coefficients obtained for different values of $k$ are shown in Table 7. If we plot coefficients $K_{k}$, $L_{k}$ and $M_{k}$ versus $k$ (shown in Fig. 10), it can be seen that the best fit curves correspond to quartic/biquadratic regression (${R^{2}\approx 0.99}$).

Fig. 9

The same as Fig. 4, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The simulations are done for BCCA structure with $N = 128$. The best fit curves correspond to quadratic regression of the form $P_{\mathrm{max}} = K_{k} n^{2} + L_{k} n + M_{k}$, which have $R^{2}\approx 0.99$. All coefficients of equation at different values of $n$ are shown in Table 7

Fig. 10

The same as Fig. 5, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The best fit curves correspond to quartic/biquadratic regression shown in (4b)–(4d), which have $R^{2}\approx 0.99$. All coefficients of (4b)–(4d) are shown in Table 8

Table 7 All co-efficients of (4a) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

Thus coefficients are given by

$$\begin{aligned}& K_{k} = \epsilon_{1} k^{4} + \epsilon_{2} k^{3} + \epsilon_{3} k^{2} + \epsilon_{4} k + \epsilon_{5} , \end{aligned}$$

(4b)

$$\begin{aligned}& L_{k} = \zeta_{1} k^{4} + \zeta_{2} k^{3} + \zeta_{3} k^{2} + \zeta_{4} k + \zeta_{5} , \end{aligned}$$

(4c)

$$\begin{aligned}& M_{k} = \eta_{1} k^{4} + \eta_{2} k^{3} + \eta_{3} K^{2} + \eta_{4} k + \eta _{5} , \end{aligned}$$

(4d)

All coefficients of (4b)–(4d) are shown in Table 8. Thus it can be seen that $P_{\mathrm{max}}$ can be also estimated using (4a) and (4b)–(4d).

Table 8 All co-efficients of (4b)–(4d) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

3.1.3 $\lambda = 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$

We also execute the computations at three different wavelengths $\lambda = 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. We find that $P_{\mathrm{max}}$ and $k$ are correlated via a polynomial regression where the degree of regression equation is 3 for $\lambda= 0.85\mbox{ }\upmu \mbox{m}$ and 2 for $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. The results are plotted in Fig. 11 where we have also included the plots for $\lambda = 0.45\mbox{ }\upmu \mbox{m}\mbox{ and }0.65\mbox{ }\upmu \mbox{m}$. Thus it can be concluded that the degree of best-fit non-linear equation depends on the wavelength of incident radiation and it decreases with the increase of wavelengths from optical to infrared. Again, if we plot $P_{\mathrm{max}}$ versus $n$ (keeping $k$ fixed), we observe that they are correlated via a quadratic equation, and this feature is observed at all three wavelengths. We also include the plots for $\lambda = 0.45\mbox{ }\upmu \mbox{m}\mbox{ and }0.65\mbox{ }\upmu \mbox{m}$. This is an interesting outcome of our work which shows $P_{\mathrm{max}}$ and $n$ are strongly correlated to each other via quadratic equation irrespective of a wavelength chosen. The results are plotted in Fig. 12. We did not show any table for these three wavelengths.

Fig. 11

$P_{\mathrm{max}}$ is plotted against $k$ for i $n = 1.4$, ii $n = 1.6$, iii $n = 1.8$ and iv $n = 2.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for five different wavelengths $\lambda = 0.45\mbox{ }\upmu \mbox{m}, 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$. The best fit curves at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$ correspond to quartic regression whereas the curves at $\lambda = 0.65\mbox{ }\upmu \mbox{m}\mbox{ and }0.85\mbox{ }\upmu \mbox{m}$ correspond to cubic regression and curves at $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$ correspond to quadratic regression. All the curves have $R^{2}\approx 0.99$

Fig. 12

$P_{\mathrm{max}}$ is plotted against $n$ for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for five different wavelengths $\lambda = 0.45\mbox{ }\upmu \mbox{m}, 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$. The best fit curves in all four panels correspond to quadratic regression, which have $R^{2}\approx 0.99$

3.2 Correlation between $S_{11}(180^{\circ})$ and complex refractive index

3.2.1 $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We now study the dependence of $n$ on the phase function $S_{11}(180^{\circ})$ (or geometric albedo) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$. We plot $S_{11}(180^{\circ})$ against $n$ for $k = 0.001, 0.005, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1$, which is shown in Fig. 13. At a fix value of $n$, $S_{11}(180^{\circ})$ decreases as $k$ increases, but this decrease is up to $k = 0.2$ (see Fig. 13(i) and Fig. 15). $S_{11}(180^{\circ})$ starts increasing when $k \ge0.3$ (see Fig. 13(ii)). The change of $S_{11}(180^{\circ})$ with $n$ for aggregate particles is small when $k$ is low, but this change is nearly independent of $n$ when $k$ is high. We have noted that $S_{11}(180^{\circ})$ and $n$ can be fitted by quartic/biquadratic regression where $R^{2}$ for each equation is ≈0.99. Thus, the best fit equation is:

$$ S_{11}\bigl(180^{\circ}\bigr) = A'_{k} n^{4} + B'_{k} n^{3} + C'_{k} n^{2} + D'_{k} n + E'_{k}, $$

(5)

where $A'$, $B'$, $C'$, $D'$ and $E'$ are coefficients of (5). The coefficients obtained from this analysis are shown in Table 9.

Fig. 13

The phase function $S_{11}(180^{\circ})$ is plotted against $n$, for $k = 0.001, 0.005, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1$, respectively. The simulations are done for BCCA structure with $N = 128$ and $a_{m} = 0.1\mbox{ }\upmu \mbox{m}$, where $\lambda$ is fixed at 0.45 μm. The best fit curves correspond to quartic/biquadratic regression of the form $S_{11}(180^{\circ}) = A' n^{4} + B' n^{3} + C' n^{2} + D' n + E'$, where $R^{2}\approx 0.99$. All coefficients of equation at different values of $k$ are shown in Table 9

Table 9 All co-efficients of (5) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We now execute the computations for $N = 512$ (BA/BPCA). The results are shown in Fig. 14. We also find that $S_{11}(180^{\circ})$ and $n$ can be fitted by quartic/biquadratic regression which is same as that of (5). We do not show any table.

Fig. 14

Same as Fig. 13, but for BA structure with $N = 512$

We now plot $S_{11}(180^{\circ})$ against $k$ for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The plot is shown in Fig. 15. Initially, an increase in $k$ lowers $S_{11}(180^{\circ})$ whereas this is not the case with high $k$. It is to be noted that $S_{11}(180^{\circ})$ and $k$ can be fitted by sextic/hexic regression which is given by:

$$ \begin{aligned}[b]S_{11}\bigl(180^{\circ}\bigr)={}&A''_{n} k^{6} + B''_{n} k^{5} + C''_{n} k^{4} + D''_{n} k^{3} \\ &+E''_{n} k^{2} + F''_{n} k + G''_{n}, \end{aligned}$$

(6)

where $A''_{n}$, $B''_{n}$, $C''_{n}$, $D''_{n}$, $E''_{n}$, $F''_{n}$ and $G''_{n}$ are coefficients of (6). The coefficients obtained from this analysis are shown in Table 10.

Fig. 15

The phase function $S_{11}(180^{\circ})$ is plotted against $k$, for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The simulations are done for BCCA structure with $N = 128$ and $a_{m} = 0.1\mbox{ }\upmu \mbox{m}$, where $\lambda$ is fixed at 0.45 μm. The best fit curves correspond to sextic/hexic regression which have $R^{2}\approx 0.99$. All coefficients of equation at different values of $k$ are shown in Table 10

Table 10 All co-efficients of (6) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We now plot the results for $N = 512$ (BA/BPCA) and is shown in Fig. 16. It is to be noted that $S_{11}(180^{\circ})$ and $k$ can be fitted by sextic/hexic regression. We do not show any table.

Fig. 16

Same as Fig. 15, but for BA structure with $N = 512$

3.2.2 $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

We now study the dependence of $n$ on the phase function $S_{11}(180^{\circ})$ at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$. We plot $S_{11}(180^{\circ})$ against $n$ for $k = 0.001, 0.005, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1$, which is shown in Fig. 17. The nature of figure is same as that of Fig. 13, but $S_{11}(180^{\circ})$ and $n$ is fitted by cubic regression ($R^{2}\approx 0.99$). The best fit equation is:

$$ S_{11}\bigl(180^{\circ}\bigr) = A'_{k} n^{3} + B'_{k} n^{2} + C'_{k} n + D'_{k}, $$

(7)

where $A'$, $B'$, $C'$ and $D'$ are coefficients of (7). The coefficients obtained from this analysis are shown in Table 11.

Fig. 17

The same as Fig. 13, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The simulations are done for BCCA structure with $N = 128$. The best fit curves correspond to cubic regression shown in (7), which have $R^{2}\approx 0.99$. All coefficients of (7) are shown in Table 11

Table 11 All co-efficients of (7) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

We now plot $S_{11}(180^{\circ})$ against $k$ for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The plot is shown in Fig. 18, which is almost same as that of Fig. 15. $S_{11}(180^{\circ})$ is minimum when $k$ is between 0.4 and 0.5. In this case, $S_{11}(180^{\circ})$ and $k$ can be fitted by quartic/biquadratic regression which is given by:

$$ S_{11}\bigl(180^{\circ}\bigr) = A''_{n} k^{4} + B''_{n} k^{3} + C''_{n} k^{2} + D''_{n} k + E''_{n}, $$

(8)

where $A''_{n}$, $B''_{n}$, $C''_{n}$, $D''_{n}$ and $E''_{n}$ are coefficients of (8). The coefficients obtained from this analysis are shown in Table 12.

Fig. 18

The same as Fig. 15, but at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The simulations are done for BCCA structure with $N = 128$. The best fit curves correspond to quartic regression shown in (8), which have $R^{2}\approx 0.99$. All coefficients of (8) are shown in Table 12

Table 12 All co-efficients of (8) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

3.2.3 $\lambda = 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$

We also study the dependence of real and imaginary part of the refractive index $(n, k)$ on $S_{11}(180^{\circ})$ for $\lambda = 0.85\mbox{ }\upmu \mbox{m}$, $1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. We find that $S_{11}(180^{\circ})$ and $k$ are correlated via cubic regression at $\lambda = 0.85\mbox{ }\upmu \mbox{m}$ and quadratic regression at $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. The results are plotted in Fig. 19 where we have also included the plots for $\lambda = 0.45\mbox{ }\upmu \mbox{m}\mbox{ and }0.65\mbox{ }\upmu \mbox{m}$. Here, we also notice that the degree of polynomial equation depends on wavelength-higher the wavelength lower is the degree of regression. Now we plot $S_{11}(180^{\circ})$ versus $n$ (keeping $k$ fixed) which is shown in Fig. 20. We observe that $S_{11}(180^{\circ})$ and $n$ are also correlated via cubic regression at $\lambda = 0.85\mbox{ }\upmu \mbox{m}$ and quadratic regression at $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. The results which are obtained for $\lambda = 0.45\mbox{ }\upmu \mbox{m}\mbox{ and }0.65\mbox{ }\upmu \mbox{m}$ are also plotted to compare with higher wavelength.

Fig. 19

$S_{11}(180^{\circ})$ is plotted against $k$ for i $n = 1.4$, ii $n = 1.6$, iii $n = 1.8$ and iv $n = 2.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for five different wavelengths $\lambda = 0.45\mbox{ }\upmu \mbox{m}, 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$. The best fit curves at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$ correspond to hexic regression whereas the curves at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$ correspond to quartic regression, curves at 0.85 μm correspond to cubic regression and curves at $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$ correspond to quadratic regression. All the curves have $R^{2}\approx 0.99$

Fig. 20

$S_{11}(180^{\circ})$ is plotted against $n$ for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for five different wavelengths $\lambda = 0.45\mbox{ }\upmu \mbox{m}, 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$. The best fit curves at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$ correspond to quartic regression whereas the curves at $\lambda = 0.65\mbox{ }\upmu \mbox{m}\mbox{ and }0.85\mbox{ }\upmu \mbox{m}$ correspond to cubic regression and curves at $\lambda = 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$ correspond to quadratic regression. All the curves have $R^{2}\approx 0.99$

3.3 Correlation between $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$

3.3.1 $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We now study the dependence of polarization maximum ($P_{\mathrm{max}}$) on phase function $S_{11}(180^{\circ})$ at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$. We have plotted $S_{11}(180^{\circ})$ versus $P_{\mathrm{max}}$ for $k = 0.001, 0.005, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8\mbox{ and }1.0$ respectively. The plot is shown in Fig. 21. At a fixed value of $k$, each point of $P_{\mathrm{max}}$ corresponds to a particular value of $n$, where extreme left point of $P_{\mathrm{max}}$-axis corresponds to $n = 2.0$ and other six points are decreasing value of $n$ from 1.9 to 1.4. We find that with increase of $P_{\mathrm{max}}$, $S_{11}(180^{\circ})$ decreases initially and reaches minimum and again increases. It then reaches maximum and decreases again. This nature is noticed when $k$ is between 0.001 and 0.1. For $k \ge0.2$, the change of $S_{11}(180^{\circ})$ is small. For low values of $k$ ($\le0.01$), if we fix the value of $P_{\mathrm{max}}$ at some point, the variation of $S_{11}(180^{\circ})$ with $k$ is found to be small. On the other hand, for higher values of $k$ (i.e., $k > 0.2$), the change of $S_{11}(180^{\circ})$ with $P_{\mathrm{max}}$ is low. It is interesting to observe that $S_{11}(180^{\circ})$ initially decreases with increase of $k$. This decrease is up to $k = 0.2$ (see Fig. 21(i)). Then $S_{11}(180^{\circ})$ starts increasing from $k = 0.3$ (see Fig. 21(ii)). The horizontal range of $P_{\mathrm{max}}$ decreases when $k$ increases. It is maximum at $k = 0.001$ (where $P_{\mathrm{max}} = [0.068, 0.845]$) and is minimum at $k = 1.0$ (where $P_{\mathrm{max}} = [0.571, 0.746]$).

Fig. 21

The phase function $S_{11}(180^{\circ})$ is plotted against the polarization maximum ($P_{\mathrm{max}}$) for the imaginary part of refractive indices ($k$) = 0.001, 0.005, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.65, 0.8 and 1.0 respectively. The simulations are done for BCCA structure with $N = 128$ and $a_{m} = 0.1\mbox{ }\upmu \mbox{m}$, where $\lambda$ is fixed at $0.45\mbox{ }\upmu \mbox{m}$. It is to be noted that for a particular value of $k$, each point of $P_{\mathrm{max}}$ corresponds to a particular value of the real part of refractive index ($n$), where extreme left point corresponds to $n = 2.0$ and other six points are decreasing value of $n$ from 1.9 to 1.4. The best fit curves correspond to cubic regression of the form $S_{11} (180^{\circ}) = W_{k} P_{\mathrm{max}}^{3} + X_{k} P_{\mathrm{max}}^{2} + Y_{k} P_{\mathrm{max}} + Z_{k}$, which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of (9a) are shown in Table 13

We find that the $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$ can be correlated via cubic regression of the form:

$$ S_{11}\bigl(180^{\circ}\bigr) = W_{k} P_{\mathrm{max}}^{3} + X_{k} P_{\mathrm{max}}^{2} + Y_{k} P_{\mathrm{max}} + Z_{k}, $$

(9a)

where $W_{k}$, $X_{k}$, $Y_{k}$ and $Z_{k}$ are $k$-dependent coefficients of (9a). The coefficient of determination ($R^{2}$) of each equation is ≈0.99. All coefficients of (9a) are now shown in Table 13. If we plot each coefficients versus $k$ (shown in Fig. 22), we find that the best fit curves correspond to quintic regression ($R^{2}\approx 0.99$).

Fig. 22

Coefficients $W_{k}$, $X_{k}$, $Y_{k}$ and $Z_{k}$ are plotted against imaginary part of the refractive index ($k$) (see Table 13). The best fit curves correspond to quintic regression shown in (9b)–(9e), which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of (9b)–(9e) are shown in Table 14

Table 13 All co-efficients of (9a) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

The coefficients are related with $k$ via the following relations:

$$\begin{aligned}& W_{k} = \alpha^{\prime}_{1} k^{5} + \alpha^{\prime}_{2} k^{4} + \alpha^{\prime}_{3} k^{3} + \alpha^{\prime}_{4} k^{2} + \alpha^{\prime}_{5} k + \alpha^{\prime}_{6} , \end{aligned}$$

(9b)

$$\begin{aligned}& X_{k} = \beta^{\prime}_{1} k^{5} + \beta^{\prime}_{2} k^{4} + \beta ^{\prime}_{3} k^{3} + \beta^{\prime}_{4} k^{2} + \beta^{\prime}_{5} k + \beta^{\prime}_{6} , \end{aligned}$$

(9c)

$$\begin{aligned}& Y_{k} = \gamma^{\prime}_{1} k^{5} + \gamma^{\prime}_{2} k^{4} + \gamma ^{\prime}_{3} k^{3} + \gamma^{\prime}_{4} k^{2} + \gamma^{\prime}_{5} k + \gamma^{\prime}_{6} , \end{aligned}$$

(9d)

$$\begin{aligned}& Z_{k} = \delta^{\prime}_{1} k^{5} + \delta^{\prime}_{2} k^{4} + \delta ^{\prime}_{3} k^{3} + \delta^{\prime}_{4} k^{2} + \delta^{\prime}_{5} k + \delta^{\prime}_{6} . \end{aligned}$$

(9e)

All coefficients of (9b)–(9e) are shown in Table 14. Using relations (1a) or (2a) and (9a) it is possible to calculate $P_{\mathrm{max}}$ and $S_{11}(180^{\circ})$ simultaneously, by knowing only $n$ and $k$ at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$.

Table 14 All co-efficients of (9b)–(9e) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

We also plot the results for BA/BPCA ($N = 512$) and is shown in Fig. 23.

Fig. 23

Same as Fig. 21, but for BA structure with $N = 512$

3.3.2 $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

At $\lambda = 0.65\mbox{ }\upmu \mbox{m}$, we find that $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$ are correlated via quadratic regression. In this case, $S_{11}(180^{\circ})$ initially decreases with increase of $k$. This decrease is up to $k = 0.5$ (see Fig. 24(i)). Then $S_{11}(180^{\circ})$ starts increasing when $k > 0.5$ (see Fig. 24(ii)). The horizontal range of $P_{\mathrm{max}}$ also decreases with increase of $k$ like that of Fig. 21. It is also observed that $S_{11}(180^{\circ})$ is almost constant at high $k$. The correlation between $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$ can be written in the form:

$$ S_{11}\bigl(180^{\circ}\bigr) = W_{k} P_{\mathrm{max}}^{2} + X_{k} P_{\mathrm{max}} + Y_{k}, $$

(10a)

where $W_{k}$, $X_{k}$ and $Y_{k}$ are $k$-dependent coefficients of (10a). The coefficient of determination ($R^{2}$) of each equation is ≈0.99. All coefficients of (10a) are now shown in Table 15. If we plot each coefficients versus $k$ (shown in Fig. 25), we find that the best fit curves correspond to cubic regression, which have coefficient of determination ($R^{2}$) ≈ 0.99.

Fig. 24

The same as Fig. 21, at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$. The simulations are done for BCCA structure with $N = 128$. The best fit curves correspond to quadratic regression of the form $S_{11} (180^{\circ}) = W_{k} P_{\mathrm{max}}^{2} + X_{k} P_{\mathrm{max}} + Y_{k}$, which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of (4a) are shown in Table 15

Fig. 25

Coefficients $W_{k}$, $X_{k}$ and $Y_{k}$ are plotted against $k$ (see Table 15). The best fit curves correspond to cubic regression shown in (10b)–(10d), which have coefficient of determination ($R^{2}$) ≈ 0.99. All coefficients of (10b)–(10d) are shown in Table 16

Table 15 All co-efficients of (10a) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

The coefficients are related with $k$ via the following relations:

$$\begin{aligned}& W_{k} = \alpha^{\prime}_{1} k^{3} + \alpha^{\prime}_{2} k^{2} + \alpha^{\prime}_{3} k + \alpha^{\prime}_{4} , \end{aligned}$$

(10b)

$$\begin{aligned}& X_{k} = \beta^{\prime}_{1} k^{3} + \beta^{\prime}_{2} k^{2} + \beta ^{\prime}_{3} k + \beta^{\prime}_{4} , \end{aligned}$$

(10c)

$$\begin{aligned}& Y_{k} = \gamma^{\prime}_{1} k^{3} + \gamma^{\prime}_{2} k^{2} + \gamma ^{\prime}_{3} k + \gamma^{\prime}_{4} . \end{aligned}$$

(10d)

All coefficients of (10b)–(10d) are shown in Table 16. Using relations (3a) or (4a) and (10a) it is possible to estimate $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$ simultaneously, by knowing $n$ and $k$ at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$.

Table 16 All co-efficients of (10b)–(10d) at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$

3.3.3 $\lambda = 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$

Now computations are executed with $\lambda = 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}$ and $1.25\mbox{ }\upmu \mbox{m}$ to study the correlation between $S_{11}(180^{\circ})$ and polarization maximum. The results obtained from simulations are plotted in Fig. 26 where the results for 0.45 μm and 0.65 μm are also included. It is interesting to note that they are also correlated via quadratic regression at three wavelengths, similar to 0.65 μm.

Fig. 26

$S_{11}(180^{\circ})$ is plotted against $P_{\mathrm{max}}$ for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for five different wavelengths $\lambda = 0.45\mbox{ }\upmu \mbox{m}, 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and } 1.25\mbox{ }\upmu \mbox{m}$. The best fit curves at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$ correspond to cubic regression whereas the curves at other four wavelengths correspond to quadratic regression. All the curves have $R^{2}\approx 0.99$

3.4 Correlation between $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$

We have investigated that the amplitude of negative polarization ($P_{\mathrm{min}}$) is prominent at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ as compared to other higher wavelengths when $N = 128$ (BCCA). So our computations are restricted to $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ only. In this case, characteristic size of aggregate ($R$) is higher than the wavelength of incident radiation when $\lambda= 0.45\mbox{ }\upmu \mbox{m}$. We now execute the code with $n = [1.4,2.0]$ and $k = [0.001,1.0]$. We have observed that the magnitude of negative polarization is less prominent (${<} 10^{-3}$) when $k > 0.1$, so we do not consider results for high $k$.

We may study the correlation between $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ keeping either $n$ or $k$ fixed. In Fig. 27, we show the effect of complex refractive indices $(n,k)$ on both $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$. The degree of negative polarization is found to be high at lower values of $k$ and is maximum at $k = 0.001$. The magnitude of $P_{\mathrm{min}}$ initially increases with increase of $P_{\mathrm{max}}$ (i.e., with decrease of $n$ from 2.0 to 1.4), which reaches maximum at some value of $P_{\mathrm{max}}$ and then decreases. Negative polarization is significantly less when $k > 0.1$. It can be seen from Fig. 27 that $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ can be fitted well via a quartic regression, if we fix $k$ and change $n$ from 2.0 to 1.4 (shown by red curves in Fig. 27).

Fig. 27

$P_{\mathrm{min}}$ is plotted against $P_{\mathrm{max}}$ for BCCA ($N=128$) particles at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$. The blue line corresponds to linear regression when $n$ is considered to be fixed, whereas the red curve represents quartic regression when $k$ is considered to be fixed

If we keep $n$ fix and change values of $k$ from 0.001 to 0.1, we observe that a strong linear correlation between $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ exists which can be written via a correlation equation:

$$ P_{\mathrm{min}} = R_{n} P_{\mathrm{max}} + S_{n}, $$

(11a)

where $R_{n}$ and $S_{n}$ are coefficients of (11a) and their values are presented in Table 17. In Fig. 28, the coefficients are plotted against $n$. The best fit curves correspond to quartic regression of the form:

$$\begin{aligned}& R_{n} = \nu_{1} n^{4} + \nu_{2} n^{3} + \nu_{3} n^{2} + \nu_{4} n + \nu_{5}, \end{aligned}$$

(11b)

$$\begin{aligned}& S_{n} = \omega_{1} n^{4} + \omega_{2} n^{3} + \omega_{3} n^{2} + \omega_{4} n + \omega_{5}. \end{aligned}$$

(11c)

All coefficients are now presented in Table 18.

Fig. 28

Coefficients $R_{n}$ and $S_{n}$ of (11a) are plotted against real part of the refractive index ($n$) (see Table 17). The best fit curves correspond to quartic regression shown in (11b)–(11c), which have $R^{2}\approx 0.99$. All coefficients of (11b)–(11c) are shown in Table 18

Table 17 All co-efficients of (11a) at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$

Table 18 All co-efficients of (11b)–(11c)

We execute the computations for $N = 512$ (BA/BPCA) at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$. The simulated data are plotted in Fig. 29 where we have included the results for $k = 0.2$. In this case, magnitude of negative polarization is ${<} 10^{-3}$ for $k > 0.2$. We find that the results are similar to that of Fig. 27 which suggests that the correlation between $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ also show the similar nature even at higher size of aggregates. We did not show any equation or table in this case.

Fig. 29

Same as Fig. 27, but for BA structure with $N = 512$

3.5 Correlation between polarization maximum and wavelength

We now study the effect of wavelength of incident radiation ($\lambda$) on polarization maximum ($P_{\mathrm{max}}$). In this work, a wide range of wavelengths 0.45, 0.55, 0.65, 0.85, 1.05, 1.15 and 1.25 μm, from optical to near-infrared, is considered. At a fixed value of $k$, $P_{\mathrm{max}}$ is plotted against $\lambda$, for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. Here we show the plots for four different values of $k$, i.e., 0.001, 0.05, 0.5 and 1.0, which are shown in Fig. 30(i), (ii), (iii) and (iv). At a fixed value of $k$, $P_{\mathrm{max}}$ increases with wavelength, which can be fitted by quartic regression equation. The vertical range of $P_{\mathrm{max}}$ is maximum at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$, where characteristic size of the aggregates ($R \approx1.1 \mbox{ }\upmu \mbox{m}$) is greater than $\lambda$. The effect of light scattering is more prominent when $R \ge\lambda$. At lower $n$, the variation of $P_{\mathrm{max}}$ with $\lambda$ is small. We did not show any equation or table in this case.

Fig. 30

$P_{\mathrm{max}}$ is plotted against $\lambda$ for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The best fit curves in all four panels correspond to quartic regression, which have $R^{2}\approx 0.99$

3.6 Correlation between $S_{11}(180^{\circ})$ and wavelength

We also study the effect of $\lambda$ on $S_{11}(180^{\circ})$ where seven wavelengths from optical to infrared ($0.45\mbox{ }\upmu \mbox{m} \le\lambda\le 1.25\mbox{ }\upmu \mbox{m}$) are taken. We show the plots for four different values of $k$, i.e., 0.001, 0.05, 0.5 and 1.0, which are shown in Fig. 31(i), (ii), (iii) and (iv). It is noted that the value of $S_{11}(180^{\circ})$ fluctuates with $\lambda$ which shows maximum value at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$ for all values of $n$. At higher value of $k$, the variation of $S_{11}(180^{\circ})$ with $\lambda$ is almost same for all values of $n$. We do not find any non-linear regression relation between $S_{11}(180^{\circ})$ and $\lambda$. However, we have joined data points by lines to show the nature of variation.

Fig. 31

$S_{11}(180^{\circ})$ is plotted against $\lambda$ for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The line points are shown for $n$ = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0

3.7 Correlation between $P_{\mathrm{max}}(0.45\mbox{ }\upmu \mbox{m})$ and $P_{\mathrm{max}}(\lambda)$

We now plot $P_{\mathrm{max}}(0.45\mbox{ }\upmu \mbox{m})$ versus $P_{\mathrm{max}}(\lambda)$ (where $\lambda = 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$) for $k = 0.001, 0.05, 0.5\mbox{ and }1.0$ which are shown in Fig. 32(i), (ii), (iii) and (iv). The data points are shown for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The best fit lines in all four panels correspond to linear regression, which have $R^{2}\approx 0.99$. This is also an interesting result coming out from this study.

Fig. 32

$P_{\mathrm{max}}(0.45\mbox{ }\upmu \mbox{m})$ is plotted against $P_{\mathrm{max}}(\lambda)$ (where $\lambda = 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$) for i $k = 0.001$, ii $k = 0.05$, iii $k = 0.5$ and iv $k = 1.0$. The simulations are done for BCCA structure with $N = 128$. The data points are shown for $n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9\mbox{ and }2.0$. The best fit lines in all four panels correspond to linear regression, which have $R^{2}\approx 0.99$

4 Results from correlation equations

In the previous sections, we have obtained the correlation equations (1a)–(11a) which can be used to calculate different scattering parameters with a given value of $n$, $k$ and $\lambda$. We now calculate $P_{\mathrm{max}}$, $S_{11}(180^{\circ})$ and $P_{\mathrm{min}}$ for BCCA structure ($N = 128$) for selected values of $n$, $k$ and $\lambda$ ($= 0.45\mbox{ }\upmu \mbox{m}$) using correlation equations and compare these with the computed values obtained using the Superposition T-matrix code. The values are displayed in Table 19. It can be seen that the values obtained from correlation equations match well with the computed values. This study shows that it is possible to study the light scattering properties of dust aggregates from the knowledge of correlation equations.

Table 19 $P_{\mathrm{max}}$, $S_{11}(180^{\circ})$ and $P_{\mathrm{min}}$ for selected values of $n$ and $k$ from computations and correlation equations at $\lambda = 0.45\mbox{ }\upmu \mbox{m}$ for BCCA structure with $N = 128$

One can also use the set of relations with appropriate conditions to model the experimental light scattering data measured by Volten et al. (2007) for fluffy aggregates of magnesiosilica and ferrosilica. The parameters $P_{\mathrm{max}}$, $A$ and $P_{\mathrm{min}}$ are important in the photopolarimetric study of comets. It has been reported from observations that comets show low geometric albedo ($A$) ∼ 4–5%, bell-shaped positive polarization branch with $P_{\mathrm{max}} \approx15\mbox{--}25\%$ at the phase angle in the range $90\mbox{--}100^{\circ}$ and negative branch of polarization with $P_{\mathrm{min}} \approx2\%$ for phase angles $\le20^{\circ}$ (Kolokolova et al. 2004). The set of relations can also be used to estimate parameters ($P_{\mathrm{max}}$, $A$ and $P_{\mathrm{min}}$) which are important for modeling the photopolarimetric properties of comet dust aggregates. But it is also important to note that the typical size of comet dust aggregates are micron sized and particles as large as 100 μm are also detected in comet 67P/Churyumov-Gerasimenko (Rotundi et al. 2015). Further, the COmetary Secondary Ion Mass Analyser (COSIMA) onboard Rosetta collected fluffy agglomerates with characteristic radius $\ge 25\mbox{ }\upmu \mbox{m}$ and porosity $> 0.5$ from the comet (Schulz et al. 2015). In this study, the simulation is mainly executed for small aggregates, and it can be extended for the higher size of the aggregates to study the light scattering properties of comet dust. This study is mainly concentrated on whether correlations exist or not among different scattering parameters in an aggregate dust model. Since a promising result has been observed in this study, so we also plan to extend our study for large aggregates in future.

5 Conclusions

• 1. At a fixed value of real part of the refractive index ($n$), the polarization maximum ($P_{\mathrm{max}}$) and imaginary part of the refractive index ($k$) are correlated via a polynomial regression. The degree of the regression depends on wavelength, higher the wavelength lower is the degree. It is found that the degree is 4 when $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ whereas it is 3 for $\lambda = 0.65\mbox{ }\upmu \mbox{m}\mbox{ and }0.85\mbox{ }\upmu \mbox{m}$, and 2 at $\lambda= 1.05\mbox{ }\upmu \mbox{m}$ and $1.25\mbox{ }\upmu \mbox{m}$. The variation of $P_{\mathrm{max}}$ is almost independent at higher $k$.

  At a fixed value of $k$, $P_{\mathrm{max}}$ and $n$ are correlated via quadratic regression which is observed at all wavelengths (optical to infra red). With increase of $n$, $P_{\mathrm{max}}$ decreases and this tendency is less prominent when $k$ is high.

• 2. At a fixed value of $n$, $S_{11}(180^{\circ})$ (or geometric albedo) and $k$ are correlated via polynomial regression where the degree of regression is found to be wavelength dependent. The degree is highest at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ and then decreases when wavelength increases. The degree of equation is found to be 6 at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$, 4 at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$, 3 at 0.85 μm, and 2 at $\lambda= 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. $S_{11}(180^{\circ})$ initially decreases with the increase of $k$ and reaches minimum and then again increases.

  When $k$ is taken to be fixed, $S_{11}(180^{\circ})$ is correlated with $n$ via a polynomial regression where the degree of regression equation depends on wavelength. The degree is found to be 4 at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ whereas it is 3 for $\lambda= 0.65\mbox{ }\upmu \mbox{m}\mbox{ and }0.85\mbox{ }\upmu \mbox{m}$, and 2 at $\lambda= 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. The change of $S_{11}(180^{\circ})$ with $n$ for aggregate particles is prominent when $k$ is low, but it is nearly independent of $n$ when $k$ is high.

• 3. When $k$ is fixed at some value, $S_{11}(180^{\circ})$ and $P_{\mathrm{max}}$ are correlated via a cubic regression at $\lambda= 0.45\mbox{ }\upmu \mbox{m}$ whereas this correlation is quadratic at other four wavelengths. We have found that $S_{11}(180^{\circ})$ decreases with increase of $P_{\mathrm{max}}$ at low $k$, but the variation is almost independent when $k$ is high.

• 4. The negative polarization is found to be high at lower values of $k$ and is maximum at $k = 0.001$. The magnitude of $P_{\mathrm{min}}$ initially increases with increase of $P_{\mathrm{max}}$ (i.e., with decrease of $n$ from 2.0 to 1.4), which reaches maximum at some value of $P_{\mathrm{max}}$ and then decreases. $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ are correlated via a quartic regression when $n$ is changed from 2.0 to 1.4 and $k$ is fixed at some value. However a strong linear correlation between $P_{\mathrm{min}}$ and $P_{\mathrm{max}}$ is noticed when $k$ is changed from 0.001 to 0.1 and $n$ is fixed at some value (for BCCA cluster with $N = 128$). However, the similar feature is also observed for comparatively larger aggregates (BA cluster with $N = 512$).

• 5. $P_{\mathrm{max}}$ increases with $\lambda$ and they are correlated via a quartic regression.

• 6. We do not find any non-linear regression relation between $S_{11}(180^{\circ})$ and $\lambda$. However, it is observed that $S_{11}(180^{\circ})$ fluctuates with $\lambda$ which shows maximum value at $\lambda= 0.65\mbox{ }\upmu \mbox{m}$.

• 7. $P_{\mathrm{max}}(0.45\mbox{ }\upmu \mbox{m})$ is linearly correlated with $P_{\mathrm{max}}(\lambda )$, where $\lambda = 0.65\mbox{ }\upmu \mbox{m}, 0.85\mbox{ }\upmu \mbox{m}, 1.05\mbox{ }\upmu \mbox{m}\mbox{ and }1.25\mbox{ }\upmu \mbox{m}$. This is an interesting result emerging out from this study. The slope of all equations is positive and this slope is highest at $\lambda = 0.65\mbox{ }\upmu \mbox{m}$.